The least common multiple of two numbers is 3780, and the greatest common divisor is 18. given that one of the numbers is 180, what is the other number?

Accepted Solution

To solve these GCF and LCM problems, factor the numbers you're working with into primes:Β 
3780 = 2*2*3*3*3*5*7
180 = 2*2*3*3*5Β 
We know that the LCM of the two numbers, call them A and B, = 3780 and that A = 180. The greatest common factor of 180 and B = 18 so B has factors 2*3*3 in common with 180 but no other factors in common with 180. So, B has no more 2's and no 5's
Now, LCM(180,B) = 3780. So, A or B must have each of the factors of 3780. B needs another factor of 3 and a factor of 7 since LCM(A,B) needs for either A or B to have a factor of 2*2, which A has, and a factor of 3*3*3, which B will have with another factor of 3, and a factor of 7, which B will have.Β 
So, B = 2*3*3*3*7 = 378.